Directivity of a distribution of baffled pistons with arbitrary phases

Using the first product theorem for arrays¹ or the bridge theorem², the directivity pattern of an arrangement of sources can be constructed by multiplying the directivity of one single source by a point source in the Fourier space.

Single baffled piston

A piston with radius $a$ is placed on a baffled plane

The directivity of the baffled piston is \begin{equation}\large D_p(\theta) = \frac{2J_1(ka\sin\theta)}{ka\sin\theta} \end{equation} and for different values of $ka$

Two pistons vibrating in phase and in anti-phase

The directivity for two pistons with relatively phase $\Phi_{12}$ is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 + e^{i(kd\sin\theta\cos\phi+\Phi_{12})}\right] \end{equation}

In phase

The directivity for when $\Phi_{12} =0$ is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 + e^{ikd\sin\theta\cos\phi}\right] \end{equation} and for different values of $ka$

In anti-phase

The directivity for when $\Phi_{12} =\pi$ is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 - e^{ikd\sin\theta\cos\phi}\right] \end{equation} and for different values of $ka$

Three Pistons on the plane in line

In this case, 2 pistons are vibrating with the same phase and the third in phase oppossition.

The directivity is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[-1 + e^{ikd\alpha} - e^{i2kd\alpha}\right] \end{equation} where $\alpha = \sin\theta\cos\phi$. For different values of $ka$

Three Pistons on the plane in "L"

In this case, 2 pistons are vibrating with the same phase and the third in phase-opposition.

The directivity is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 - e^{ikd\alpha} - e^{ikd\beta}\right] \end{equation} where $\alpha = \sin\theta\cos\phi$ and $\beta = \sin\theta\sin\phi$. For different values of $ka$

Four pistons

Here, 2 pistons are vibrating with the same phase and the other two in phase opposition.

The directivity is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[-1 + e^{ikd\alpha} - e^{ik2d\alpha} + e^{ik3d\alpha}\right] \end{equation} where $\alpha = \sin\theta\cos\phi$. For different values of $ka$

Four pistons forming a squared

Again, two pistons are vibrating with the same phase and the other two in phase opposition.

\begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 - e^{ikd\alpha} - e^{ikd\beta} + e^{i(kd\alpha + kd\beta)} \right] \end{equation}

where $\alpha = \sin\theta\cos\phi$ and $\beta = \sin\theta\sin\phi$. For different values of $ka$

References

[1] E. G. Williams, FOURIER ACOUSTICS Sound Radiation and Nearfield Acoustical Holography. 1999.

[2]L. L. Beranek and T. J. Mellow, Acoustics Sound Fields, Transducers and Vibration, Second edition. Academic Press, 2019. doi: 10.1016/B978-0-12-391421-7.00013-0.

Please refer to the paper : García A et al,Radiation efficiency of a distribution of baffled pistons with arbitrary phases, JASA, 2022 for more information.